Harmonic Division

Harmonic Division∗ Russelle Guadalupe May 10, 2011

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Definition

Let A, B, C, D be four collinear points in this order. The four-point (ACBD) is harmonic division or harmonic if and only if they satisfy AD AC = .

CB DB If P is a point not collinear with A, B, C, D, we define a pencil P (ABCD) to be the four lines P A, P B, P C, P D. P (ACBD) is harmonic when (ACBD) is harmonic.

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Theorems 1. If the division (ACBD) is harmonic and O is the midpoint of AB, then OB 2 = OC · OD. 2. If AX, BY, CZ are three concurrent cevians of triangle ABC, with X, Y, Z lying on sides BC, CA, AB respectively, and if ZY ∩ BC = T , then (BXCT ) is harmonic.

3. If four concurrent lines are such that one transversal cuts them harmonically, then every transversal intersects these four lines to form a harmonic division.

4. Let A, B, C, D be four collinear points in this order. If P is a point not collinear with A, B, C, D, then any of the two statements imply the third: a) The division P (ACBD) is harmonic. b) P B bisects ∠CP D internally. c) AP ⊥ P B. 5. If (ACBD) and (A0 C 0 B 0 D0 ) are harmonic, and AA0 , BB 0 , CC 0 concur at P , then D, D0 , P are collinear.

6. Let ABCD be a convex quadrilateral. If K = AD ∩ BC, M = AC ∩ BD, P = AB ∩ KM and Q = DC ∩ KM , then (KM P Q) harmonic. ∗ This

is an unedited version; some theorems are stated without proofs.

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7. Let M be the midpoint of segment AB, and P∞ be the point of infinity on the line AB. Then, (AM BP∞ ) is harmonic.

8. P C, P D are tangents from an external point P to the circle ω. A line through P meets ω at A, B (so P, A, B collinear in this order). If AC ∩ BD = Q, then ACBD is a harmonic quadrilateral and (P AQB) is harmonic.

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Problems 1. AD, BE, CF are the altitudes of triangle ABC. If DE ∩ AB = F 0 and EF ∩ BC = D0 , prove that F D and F 0 D0 meet on AC. 2. The tangent to the circle ω with diameter AB at the external point P meets ω at T , and N ∈ AB such that P N ⊥ AB. The line joining B and the midpoint of T P meets P N at Q. Prove that AQ k T P .

3. ABC is a triangle and the circle ω is tangent to the sides AB, AC and intersects the side BC in two distinct points. Let one of these two points be K. Let P, Q be on the line BC and be outside of the segment BC such that KB = P B and KC = QC. Let AK intersect ω [in 4ABC] at M .

Prove that the circumcircle of 4P M Q is tangent to ω.

4. Let 4ABC be an acute triangle, and let D be the projection of A on BC. Let M, N be the midpoints of AB and AC respectively. Let Γ1 and Γ2 be the circumcircles of 4BDM and 4CDN respectively, and let K be the other intersection point of Γ1 and Γ2 . Let P be an arbitrary point on BC and E, F are on AC and AB respectively such that P EAF is a parallelogram. Prove that if M N is a common tangent line of Γ1 and Γ2 , then K, E, A, F are concyclic.

5. Let ABCD be a convex quadrilateral with A, B, C, D concyclic. Assume DA AB = . Let Γ be a circle through A and D, ∠ADC is acute and BC CD tangent to AB, and let E be a point on Γ and inside ABCD. Prove that AE ED AE ⊥ EC if and only if − = 1.

AB AD 6. The incircle with center I of 4ABC touches it in the points M ∈ BC , N ∈ CA. Denote the midpoints E and F of AB and AC respectively.

Prove that the lines M N, EF, BI are concurrent.

7. The circle Γ is inscribed to the scalene triangle ABC. Γ is tangent to the sides BC, CA and AB at D, E and F respectively. The line EF intersects the line BC at G. The circle of diameter GD intersects Γ in R (R 6= D).

Let P, Q(P 6= R, Q 6= R) be the intersections of Γ with BR and CR, respectively. The lines BQ and CP intersect at X. The circumcircle of CDE meets QR at M , and the circumcircle of BDF meet P R at N .

Prove that P M, QN and RX are concurrent.

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